SCATTERING OF PHOTONS OF VARIOUS ENERGIES BY ELECTRONS

L. V. Kurnosova

CONTENTS

Introduction . . . 603

1. Conservation laws for energy and momentum in Compton scattering . . . 605

2. Effective cross section of Compton scattering by free electrons . . . 608
   a) Differential and integral scattering cross sections on a free electron at rest . . . 608
   b) Cross section of Compton scattering by a moving electron . . . 614
   c) Polarization in the Compton effect on oriented electrons . . . 619
   d) Results of the new quantum electrodynamics . . . 623

3. Compton scattering on various elementary particles . . . 625
   a) Summary of effective cross sections for photon scattering by charged particles possessing various spins and magnetic moments . . . 625
   b) Scattering of photons by nucleons . . . 626

4. Experimental data at low energies . . . 627

5. Experimental data at high energies . . . 639

INTRODUCTION

The question of the scattering of photons by electrons, as is well known, played a significant role in the development of quantum concepts. In 1923 Compton¹ observed the phenomenon of a change in the frequency of X-rays when they are scattered by electrons. Compton, and independently Debye², wrote down relations for this process based on the laws of conservation of energy and momentum, under the assumption of the existence of the photon as a particle with energy \(h\nu\) and momentum \(\frac{h\nu}{c}\).

The quantum nature of light could be established with particular clarity by observing recoil electrons in the process of scattering of light by free electrons. D. V. Skobeltsyn[^6] applied the method of observing the tracks of recoil electrons in a Wilson chamber placed in a magnetic field. This method made it possible, from the curvature of the tracks, to determine the energy of the recoil electrons and thus to establish experimentally the relation between the energy and the angle of emission of the recoil electron. D. V. Skobeltsyn’s experiments on the investigation of the angular correlations and angular distribution of recoil electrons served as confirmation of the conceptions concerning the quantum nature of γ-radiation.

Numerous investigations were devoted to clarifying the fundamental question of the validity of the laws of conservation of energy and momentum in Compton scattering. The statistical theory of scattering of Bohr, Kramers, and Slater[^3] proved to be incorrect, and the experiments of Shankland[^4], which had supposedly confirmed this theory, were erroneous. The entire subsequent development of knowledge about the scattering process confirmed the correctness of the initial point of view concerning the validity of the conservation laws. Investigations by D. V. Skobeltsyn and a number of other works[^5][^7][^8] showed that, within the limits of experimental accuracy (of the order of 10%), the angular distribution of electrons and photons (the differential cross section) agrees with the formula obtained by Klein, Nishina[^9], and the Soviet physicist I. E. Tamm[^10] on the basis of the Dirac equation.

The question of photon scattering, as one of the simplest types of interaction of elementary particles, is also of interest at the present time. In connection with the successes of the new quantum electrodynamics, developed in recent years[^11], theoretical calculations of cross sections with allowance for radiative corrections have been made for various processes, in particular also for Compton scattering.

Ever greater interest is being shown in problems of scattering of high-energy photons by protons and other particles. At high energies of the incident photons it may turn out that the cross section depends on the “structure” of the particles. Thus, for example, in the scattering of photons of sufficiently high energy by nucleons (protons or neutrons), one may expect that the scattering cross section depends on the structure of the “meson cloud” surrounding the nucleon. The character of the dependence of the cross section on the energy of the incident photons (for photon energies close to the meson-production threshold) may experimentally justify the choice among various variants of the meson theory of nuclear forces.

A more detailed study of scattering at high energies became possible thanks to the development of accelerator technology, the appearance of betatrons and synchrotrons, which provide photons with ener-

gies of the order of hundreds of mega-electron-volts. If earlier the study of Compton scattering served as a key to the development of ideas about the corpuscular nature of light, then now the study of Compton scattering by various particles may serve as a key to establishing the “structure” of particles and provide an idea of the nature of the coupling of electromagnetic and nuclear fields.

The review gives a summary of formulas describing the scattering of photons of various energies by electrons and by other particles, and describes the most important experimental work in this field.

1. LAWS OF CONSERVATION OF ENERGY AND MOMENTUM IN COMPTON SCATTERING

In the scattering of radiation with a large wavelength by free electrons, the electromagnetic theory of light is valid, and the frequency of the scattered radiation is equal to the frequency of the incident radiation. In the case of scattering of radiation with a small wavelength, i.e. with a high frequency \((h\nu \gg mc^2;\ h\) is Planck’s constant, \(m\) is the rest mass of the electron), the corpuscular properties of electromagnetic radiation appear. In this case the scattering process should be regarded as an elastic collision of two particles: an electron and a photon with energy \(h\nu\) and momentum \(\dfrac{h\nu}{c}\). If before scattering the electron was at rest, then from the laws of conservation of energy and momentum the following relations follow:

\[ \left. \begin{aligned} h\nu &= h\nu' + mc^2\left(\frac{1}{\sqrt{1-\beta^2}}-1\right),\\ \frac{h\nu}{c} &= \frac{h\nu'}{c}\cos\theta + \frac{m\beta c}{\sqrt{1-\beta^2}}\cos\varphi,\\ -\frac{h\nu'}{c}\sin\theta + \frac{m\beta c}{\sqrt{1-\beta^2}}\sin\varphi &= 0, \end{aligned} \right\} \tag{1} \]

where \(\beta=\dfrac{v}{c}\) is the ratio of the recoil-electron velocity to the speed of light*), \(\nu\) and \(\nu'\) are the frequencies of the incident and scattered photons,

*) We give the numerical values of some basic constants according to the data published in 13, 52:

\[ h=6.6242\cdot10^{-27}\ \text{erg}\cdot\text{sec},\quad m=9.1064\cdot10^{-28}\ \text{g},\quad c=2.9978\cdot10^{10}\ \frac{\text{cm}}{\text{sec}}, \]

\[ e=4.8025\cdot10^{-10}\ \mathrm{CGSE}. \]

θ and φ are, respectively, the scattering angle of the photon and the “recoil angle” of the electron (Fig. 1).

It follows, moreover, from the law of conservation of momentum that the directions of the incident and scattered photons and of the recoil electron lie in one plane. The first of relations (1) expresses the law of conservation of energy in scattering: the energy of the incident photon is equal to the sum of the energy of the scattered photon and the kinetic energy of the electron acquired by it in the act of scattering. The other two relations (1) represent the law of conservation of momentum; namely, the first of them corresponds to the conservation of the momentum component in the direction of propagation of the incident radiation, and the second to that in the direction perpendicular to the direction of propagation of the incident radiation.

By simple algebraic transformations one can obtain from (1) relations connecting the frequencies of the incident and scattered radiations:

\[ \frac{h\nu'}{mc^2}=\frac{h\nu}{mc^2+h\nu(1-\cos\theta)} \tag{2} \]

or, for the corresponding wavelengths:

\[ \lambda'=\lambda+\lambda_\theta(1-\cos\theta), \tag{3} \]

where \(\lambda_\theta=2\pi\lambda_0\), and \(\lambda_0=\dfrac{h}{2\pi mc}=3.8619\cdot10^{-11}\ \text{cm}\) is the Compton wavelength.

Let us express the kinetic energy of the electron \(\varepsilon\) by the quantity \(\gamma_2=\dfrac{\varepsilon}{mc^2}\). In these units, putting \(\dfrac{h\nu'}{mc^2}=\gamma_1,\ \dfrac{h\nu}{mc^2}=\gamma\), we rewrite the first of relations (1) in the form

\[ \gamma=\gamma_1+\gamma_2, \]

and (2) in the form

\[ \gamma_1=\frac{\gamma}{1+\gamma(1-\cos\theta)}. \tag{4} \]

Fig. 1. Compton effect. \(h\nu\) is the energy of the incident photon, \(h\nu'\) the energy of the scattered photon, \(mc^2\left(\dfrac{1}{\sqrt{1-\beta^2}}-1\right)\) the energy of the recoil electron, \(\theta\) the photon scattering angle, \(\varphi\) the electron recoil angle.

Relations (2), (3), (4) indicate that, in the scattering of photons whose energy \(h\nu \gtrsim mc^2\), the energy of the scattered photon is not equal to the energy of the incident photon, and consequently the frequencies of the incident and scattered radiations are also different: the frequency of the scattered radiation is always less than the frequency of the incident radiation (approaching it as the scattering angle decreases). In the case of a very large energy of the incident photon (\(h\nu \gg mc^2\)), the energy of the scattered photon is not

can be less than \(\dfrac{mc^2}{2}\) for any scattering angle. The laws of conservation of energy and momentum make it possible to establish relations also between other quantities characterizing the scattering. It is easy to obtain a number of relations useful for experimenters. For the energy of the recoil electron we obtain:

\[ \gamma_2=\frac{2\gamma^2}{1+2\gamma+(1+\gamma)^2\operatorname{tg}^2\varphi}, \tag{5} \]

where \(\gamma_2\) is the kinetic energy of the recoil electron, and \(\varphi\) is the angle between the direction of the incident radiation and the direction of motion of the electron after the act of scattering.

For the angle of emission of the recoil electron we have:

\[ \operatorname{ctg}\varphi=(1+\gamma)\operatorname{tg}\frac{\theta}{2}, \tag{6} \]

where \(\theta\) and \(\varphi\) have the previous meaning.

For the photon scattering angle the following relations hold:

\[ \begin{aligned} \sin^2\theta&=\left(\frac{1}{\gamma_1}-\frac{1}{\gamma}\right)\left[2-\left(\frac{1}{\gamma_1}-\frac{1}{\gamma}\right)\right],\\ \sin\theta&=\frac{2(\gamma+1)\operatorname{tg}\varphi}{1+(1+\gamma)^2\operatorname{tg}^2\varphi},\\ \cos\theta&=\frac{(1+\gamma)^2\operatorname{tg}^2\varphi-1}{(1+\gamma)^2\operatorname{tg}^2\varphi+1}. \end{aligned} \tag{7} \]

Thus, from the conservation laws one can obtain all the energy and angular relations for photon scattering by an electron and by other particles. As an example, in Table I we give the dependence of the recoil-electron energy on the photon scattering angle (for an incident-photon energy of 200 MeV).

The probabilities of photon scattering and of the emission of a recoil electron with a given energy in a given direction are obtained in quantum electrodynamics and are considered in the following paragraph.

Table I

Dependence of the recoil-electron energy on the photon scattering angle \(\theta\).
\(\gamma=400\) (energy 200 MeV)

\(\theta^\circ\)	\(\varphi\)	\(\gamma_1\)	\(\gamma_2\)	\(\theta^\circ\)	\(\varphi\)	\(\gamma_1\)	\(\gamma_2\)
0	\(90^\circ\)	400	0	60	\(0^\circ14'\)	1,9	398,1
2	\(8^\circ08'\)	370	30	90	\(0^\circ08'\)	0,9	399,1
4	\(4^\circ00'\)	200	200	105	\(0^\circ06'\)	0,7	399,3
6	\(2^\circ44'\)	125	275	120	\(0^\circ04'\)	0,6	399,4
10	\(1^\circ38'\)	56,5	343,5	150	\(0^\circ02'\)	0,5	399,5
20	\(0^\circ48'\)	15,9	384,1	165	\(0^\circ01'\)	0,5	399,5
30	\(0^\circ32'\)	7,3	392,7	180	\(0^\circ\)	0,5	399,5
45	\(0^\circ20'\)	3,4	396,6

2. EFFECTIVE CROSS SECTION OF COMPTON SCATTERING ON FREE ELECTRONS

This section gives a summary of formulas for the effective cross sections of Compton scattering on free electrons at rest and in motion. Expressions for the cross sections are considered for cases of different polarization of the incident photons and different orientation of the electrons in the scattering substance. Expressions are also given for radiative corrections to the scattering cross section, obtained recently by new methods of quantum electrodynamics.

a) Differential and integral cross sections of scattering on a free electron at rest

The scattering of light by free electrons in the classical case is described as the radiation of an electron oscillating under the action of an external light field. The effective scattering cross section for unpolarized incident radiation is expressed by Thomson’s formula:

\[ d\sigma=\frac{r_0^2}{2}(1+\cos^2\theta)\,d\Omega , \tag{8} \]

where

\[ r_0=\frac{e^2}{mc^2}=2.8182\cdot 10^{-13}\ \text{cm} \]

is the so-called “classical” electron radius, \(\theta\) is the photon scattering angle, and the integral cross section is equal to

\[ \sigma_0=\frac{8}{3}\pi r_0^2=6.6537\cdot 10^{-25}\ \text{cm}^2 . \]

The effective scattering cross section in relativistic quantum electrodynamics is obtained by the method of perturbation theory when considering the interaction of a photon with an electron, the electron being described by the Dirac equation. In doing so, an expansion of the expression for the energy of interaction of the photon with the electron is used in powers of the small quantity

\[ \frac{e^2}{\hbar c}\simeq \frac{1}{137}, \]

where \(\hbar=\frac{h}{2\pi}\). Since the matrix elements of the interaction are different from zero only for those transitions in which only one photon is absorbed or emitted, the scattering process, occurring with the participation of two photons, can take place only through intermediate states. The probability of such transitions is proportional to \(e^4\), since the probability of absorption and the probability of emission are proportional to \(e^2\). Calculation of the matrix elements of the interaction and summation over intermediate states lead to the following formula for the differential scattering cross section in the case of plane-polarized incident radiation, obtained for the first—

... by Klein–Nishina\(^9\) and, independently, more rigorously derived by I. E. Tamm\(^ {10}\):

\[ d\sigma=\frac{1}{4}r_0^2\frac{\gamma_1^2}{\gamma^2} \left[ \frac{\gamma}{\gamma_1}+\frac{\gamma_1}{\gamma}-2+4\cos^2\Theta \right]d\Omega . \tag{9} \]

Formula (9) determines the number of photons scattered into the solid angle \(d\Omega\), whose axis makes an angle \(\theta\) with the direction of the incident photons. The quantities \(\gamma, \gamma_1, r_0\) have the same meaning as before, and \(\Theta\) is the angle between the polarization directions of the incident and scattered photons (i.e., the angle between the electric vector of the incident radiation \(e_0\) and the electric vector of the scattered radiation \(e\)). If the incident radiation is unpolarized, then from formula (9), by averaging over the angle \(\alpha\) between the plane drawn through the direction of the incident beam and its polarization vector and the plane drawn through the directions of the incident and scattered beams, we obtain

\[ d\sigma=\frac{1}{2}r_0^2\frac{\gamma_1^2}{\gamma^2} \left[ \frac{\gamma}{\gamma_1}+\frac{\gamma_1}{\gamma}-\sin^2\theta \right]d\Omega . \tag{10} \]

In the case of plane-polarized incident radiation it is convenient to represent the scattered radiation as consisting of two linearly polarized components\(^ {12}\). We shall denote by \(d\sigma_\perp\) the scattering cross section in the case when the polarization vector of the scattered radiation is perpendicular to the polarization vector of the incident radiation \((\cos\Theta=0)\). We shall denote by \(d\sigma_{\parallel}\) the scattering cross section in the case when the polarization vectors of the incident and scattered radiation lie in one plane with the direction of the incident radiation. It follows directly from (9) that in the nonrelativistic case \((h\nu \ll mc^2)\), when, according to (2)—(4), the frequency of the scattered radiation is equal to the frequency of the incident radiation, the values of the cross sections \(d\sigma_\perp\) and \(d\sigma_{\parallel}\) will be as follows:

\[ d\sigma_\perp=0,\quad d\sigma_{\parallel}=r_0^2\cos^2\Theta\,d\Omega . \tag{11} \]

If the polarization vectors of the incident and scattered radiation lie in one plane with the direction of the scattered radiation, then it is easy to express the angle \(\Theta\) between the polarization vectors through the photon scattering angle \(\theta\) and the angle \(\alpha\). It is easy to show that

\[ \cos^2\Theta=1-\cos^2\alpha\sin^2\theta, \tag{12} \]

whence, instead of (11), we obtain:

\[ d\sigma_\perp=0;\quad d\sigma_{\parallel}=r_0^2\,d\Omega\,(1-\sin^2\theta\cos^2\alpha). \tag{13} \]

This means that the scattered radiation is completely polarized if the incident radiation is polarized. The result obtained...

corresponds to the classical representation of the scattering of light by a free electron oscillating under the action of the incident light wave in the direction of its electric vector. Formula (13) coincides with Thomson’s formula for polarized incident radiation. In the case of unpolarized incident radiation, from (13) we obtain (8) by averaging over \(\alpha\).

In the extreme relativistic case, when \(h\nu \gg mc^2\), one must distinguish scattering through small angles and through large angles. For scattering through sufficiently small angles, according to (4) the frequency of the scattered radiation is equal to the frequency of the incident radiation. From (4) it is seen that the condition that the frequency not change upon scattering is the inequality

\[ \gamma(1-\cos\theta)\ll 1. \tag{14} \]

In this case \(\gamma_1 \simeq \gamma\), and formulas (11), (13) for the cross section are again obtained.

The situation is quite different for scattering through large angles. Condition (14) is not satisfied in this case, and from (4) we obtain the relation

\[ \gamma_1 \simeq \frac{1}{1-\cos\theta} \tag{15} \]

(since \(\gamma \gg 1\)).

It follows from (15) that \(\gamma_1 \ll \gamma\), and therefore the Klein–Nishina–Tamm formula (10) gives, in the case of large scattering angles,

\[ d\sigma_{\perp}=d\sigma_{\parallel} = \frac{1}{4}\, r_0^2\, \frac{\gamma_1}{\gamma}\, d\Omega, \tag{16} \]

since the second and third terms standing in the brackets of formula (10) are small in comparison with the first term.

The relation obtained shows that in the extreme relativistic case the scattered radiation is not polarized \((d\sigma_{\perp}=d\sigma_{\parallel})\) even for polarized incident radiation, i.e. as a result of Compton scattering through large angles the radiation is completely depolarized. It should be noted that in the case \(h\nu \sim mc^2\), when the phenomenon is no longer described by Thomson’s classical formula, it is seen from formula (9) that

\[ d\sigma_{\parallel}>d\sigma_{\perp}, \tag{17} \]

i.e. that even in scattering through large angles the scattered radiation is partially polarized in the plane passing through the polarization vector of the incident radiation \(\mathbf e_0\) and through the direction of scattering. From relation (12) it follows that the maximum partial polarization is achieved if the polarization vector of the incident radiation is perpendicular to the plane passing through the directions of the incident and scattered rays. Therefore, in the scattering of radiation that was initially unpolarized, the scattered

the radiation is partially polarized in the direction perpendicular to the scattering plane.

The Klein–Nishina–Tamm formula can be represented in a somewhat different form if the energy of the scattered photon is expressed through its scattering angle, and the solid angle through the scattering angle \(\theta\) and the azimuthal angle \(\alpha'\):

\[ d\sigma=\frac{r_0^2}{2}\left\{ \frac{\left[\gamma(1-\cos\theta)+\cos^2\theta\right]\left[1+\gamma(1-\cos\theta)\right]+1} {\left[1+\gamma(1-\cos\theta)\right]^3} \right\}\sin\theta\,d\theta\,d\alpha'. \tag{18} \]

In some cases it is more convenient to eliminate the angle \(\theta\) from formula (10) and express the cross section as a function of the energies of the incident and scattered photons. For this purpose we note that, according to (4),

\[ d\gamma_1=-\frac{\gamma^2\sin\theta\,d\theta}{\left[1+\gamma(1-\cos\theta)\right]^2} =-\gamma_1^2\sin\theta\,d\theta, \tag{19} \]

and, consequently,

\[ |d\Omega|=|\sin\theta\,d\theta\,d\alpha'| =\left|\frac{d\gamma_1}{\gamma_1^2}\,d\alpha'\right|. \tag{20} \]

Substituting (20) into (10) and replacing \(\theta\) by \(\gamma\) and \(\gamma_1\), according to (7), we obtain:

\[ d\sigma=\frac{r_0^2}{2}\frac{d\gamma_1}{\gamma_1}\frac{1}{\gamma} \left[ 1+\left(\frac{\gamma_1}{\gamma}\right)^2 -2\frac{\gamma_2}{\gamma^2} +\frac{\gamma_2^2}{\gamma^3\gamma_1} \right]d\alpha', \tag{21} \]

where \(\gamma, \gamma_1, \gamma_2\) are the energies of the incident photon, the scattered photon, and the kinetic energy of the recoil electron, expressed in units of \(mc^2\).

For \(\gamma\gg 1\), formula (21) passes, at small \(\theta\), into

\[ d\sigma=\frac{r_0^2}{2}\frac{d\gamma_1}{\gamma_1}\frac{1}{\gamma} \left[ 1+\left(\frac{\gamma_1}{\gamma}\right)^2 \right]d\alpha'. \tag{22} \]

Replacing in (10) \(\sin\theta\) by formula (7), \(\gamma_1\) by formula (4), and expressing the solid angle \(d\Omega\) through \(\varphi,\alpha',d\varphi,d\alpha'\), we obtain, after integration over \(\alpha'\), the following formula:

\[ \begin{aligned} d\sigma={}&4\pi r_0^2 \frac{(1+\gamma)^2\tan\varphi} {\cos^2\varphi\left[1+2\gamma+(1+\gamma)^2\tan^2\varphi\right] \left[1+(1+\gamma)^2\tan^2\varphi\right]} \\ &\times \left\{ 1-\frac{2}{\gamma} -\frac{2}{\gamma^2} +\left(\frac{2}{\gamma}+\frac{1}{\gamma^2}\right) \frac{1+(1+\gamma)^2\tan^2\varphi} {1+2\gamma+(1+\gamma)^2\tan^2\varphi} \right. \\ &\left. +\frac{1}{\gamma^2} \frac{1+2\gamma+(1+\gamma)^2\tan^2\varphi} {1+(1+\gamma)^2\tan^2\varphi} +\frac{\left[1+(1+\gamma)^2\tan^2\varphi\right]^2} {\left[1+2\gamma+(1+\gamma)^2\tan^2\varphi\right]^2} \right\}d\varphi, \tag{23} \end{aligned} \]

expressing the differential cross section of Compton scattering with the electron emitted at an angle \(\varphi\) in the angular interval \(d\varphi\). This formula is more convenient for comparison with experiment than

formula (10), since it is usually precisely the recoil electrons that are observed. As an example, let us give several graphs of the differential cross section in polar coordinates. Fig. 2 shows the differential cross section per unit solid angle for electrons scattered through an angle \(\varphi\); Fig. 3 shows the same per unit angle \(\varphi\). Similar graphs are shown in Figs. 4 and 5 for the number of photons emitted in the direction \(\theta\).

Fig. 2. Graph of the differential cross section per unit solid angle for the number of electrons scattered through a given angle \(\varphi\), for various energies of the incident photons: \(1\) — for \(\gamma=1\); \(2\) — for \(\gamma=2.3\); \(3\) — for \(\gamma=5.4\).

Fig. 3. Graph of the differential cross section per unit angle \(\varphi\) for the number of electrons scattered through a given angle, for various energies of the incident photons: \(1\) — for \(\gamma=1\); \(2\) — for \(\gamma=2.3\); \(3\) — for \(\gamma=5.4\).

Fig. 4. Graph of the differential cross section per unit solid angle for the number of photons emitted in the direction \(\theta\), for various energies of the incident photons: \(1\) — for \(\gamma=0\); \(2\) — for \(\gamma=0.1\); \(3\) — for \(\gamma=0.4\); \(4\) — for \(\gamma=1\); \(5\) — for \(\gamma=4\); \(6\) — for \(\gamma=10\).

Integrating expression (18) with respect to \(\theta\) and with respect to \(\alpha'\), we obtain the integral cross section in the form

\[ \sigma = 2\pi r_0^2 \left\{ \frac{1+\gamma}{\gamma^3} \left[ \frac{2\gamma(1+\gamma)}{1+2\gamma} -\ln(1+2\gamma) \right] + \frac{1}{2\gamma}\ln(1+2\gamma) - \frac{1+3\gamma}{(1+2\gamma)^2} \right\}. \tag{24} \]

For a small energy of the incident photon \((\gamma \ll 1)\), expression (24) becomes

\[ \sigma = \frac{8\pi}{3} r_0^2 \left( 1 - 2\gamma + \frac{26}{5}\gamma^2 - \cdots \right). \tag{25} \]

For a large energy of the incident photon \((\gamma \gg 1)\) we obtain:

\[ \sigma = \pi r_0^2 \frac{1}{\gamma}\left( \ln 2\gamma + \frac{1}{2} \right). \tag{26} \]

Figure 6 shows the dependence of \(\sigma\) on the energy of the incident photon. Expression (26) and the curve in Fig. 6 show that the cross section

Fig. 5. Graph of the differential cross section per unit solid angle for the number of photons emitted in the directions \(\theta\), for various energies of the incident photons: \(1\)—for \(\gamma \approx 0\); \(2\)—for \(\gamma = 0.1\); \(3\)—for \(\gamma = 0.4\); \(4\)—for \(\gamma = 1\); \(5\)—for \(\gamma = 4\); \(6\)—for \(\gamma = 10\).

Fig. 6. Dependence of the Compton-scattering cross section on the energy of the incident photon.

of Compton scattering decreases as the energy of the incident photons increases.

Modern ideas about the electron compel us to think that the Klein–Nishina–Tamm formula is valid up to extremely large energies of the incident photons. Deviations from the usual Klein–Nishina–Tamm formula may be expected owing to the presence of the radiation-reaction force. In classical electrodynamics the Thomson formula for the total integral scattering cross section with allowance for radiation reaction has the form\({}^{14}\)

\[ \sigma = \frac{8}{3}\pi r_0^2 \left( 1 - \frac{4}{9}\frac{r_0^2}{\lambda^2}\,4\pi^2 + \cdots \right). \tag{27} \]

In the region where classical electrodynamics is in general applicable \(\left(\bar{\lambda} \gg \frac{h}{2\pi mc},\ \text{i.e.}\ \bar{\lambda} \gg \bar{\lambda}_0 = 3.8619 \cdot 10^{-11}\ \text{cm}\right)\), the corrections due to the reaction force are small.

In quantum electrodynamics, for the region of large energies of the incident photons, P. Nemirovskii\({}^{14}\) obtained the formula

\[ \sigma = \pi r_0^2 \frac{1}{\gamma}\left( \ln 2\gamma + \frac{1}{2} \right)\left[1 - \alpha^2(\ln 2\gamma)^2\right], \tag{28} \]

differing from the Klein–Nishina–Tamm formula by the factor \([1-\alpha^2(\ln 2\gamma)^2]\), where \(\alpha=\dfrac{2\pi e^2}{hc}\simeq \dfrac{1}{137}\). From (28) it is seen that the radiation reaction has a strong effect only at very high energies of the incident photons. The second term in the bracket \([1-\alpha^2(\ln 2\gamma)^2]\) becomes comparable with unity (which corresponds to a one-hundred-percent deviation from the usual formula for the integral effective cross section) only when the energy of the incident photon is \(h\nu\sim 2\cdot 10^{60}\) eV, i.e. at a wavelength \(\lambda\simeq 10^{-70}\) cm. At lower energies the corrections decrease, reaching, for \(h\nu\sim 10^{16}\) eV, approximately \(2.5\%\), and for \(h\nu\sim 137mc^2\) less than \(0.2\%\).

Thus, in quantum electrodynamics, which treats the electron as pointlike, the Klein–Nishina–Tamm formula is valid up to very high energies of the incident photons.

b) Cross Section for Compton Scattering on a Moving Electron

In certain problems, for example in considering Compton scattering of photons by cosmic-ray particles outside the Earth’s atmosphere¹⁵, it is necessary to consider scattering on a moving electron. The scattering may be considered either in a coordinate system connected with the electron, or in the laboratory frame of reference. In the latter case it is necessary to transform the effective scattering cross section in the corresponding way.

In this section a formula is given for the scattering cross section, written for such a coordinate system in which the electron moves with an arbitrary constant velocity. In particular, from this formula an expression is obtained for the scattering cross section in the center-of-inertia system of the incident photon and electron.

The differential cross section \(\sigma(\gamma,\theta)\,d\Omega\) for Compton scattering on an electron moving with constant velocity can be obtained by transforming the Klein–Nishina–Tamm formula from the frame of reference in which the electron is at rest. The scattering cross section may be defined as the ratio of the number of photons scattered into a given solid angle per unit time to the number of photons which in the same time have passed through a fixed unit area.

For convenience in transforming the scattering cross section from one coordinate system to another, let us introduce an auxiliary quantity \(A(\gamma,\theta)\,d\Omega\), defined as follows. Suppose that, out of a number of particles \(N_1\) passing through a fixed area of size \(q\), \(dN_2\) particles are scattered into the solid angle \(d\Omega\). We denote the ratio of \(dN_2\) to \(N_1/q\) by \(A(\gamma,\theta)\,d\Omega\).

Obviously, the quantities \(N_1\) and \(dN_2\) are invariant with respect to Lorentz transformations. Pauli \(^{16}\) showed that the cross section \(q\) of a photon beam is invariant with respect to Lorentz transformations*). Therefore, in the case of photon scattering the quantity

\[ A(\gamma,\theta)\,d\Omega=\frac{dN_2}{\dfrac{N_1}{q}} \tag{29} \]

is an invariant. Here \(\gamma\) is the photon energy, \(\theta\) is the scattering angle.

Thus, knowing the transformation law for the solid angle, from
\(A(\gamma,\theta)d\Omega=A_0(\gamma_0,\theta_0)d\Omega_0\) it is easy to find the quantity \(A(\gamma,\theta)\) in the new coordinate system, if \(A_0(\gamma_0,\theta_0)\) is its value in the old frame of reference. In the case of an electron at rest, the cross section
\(\sigma_0(\gamma_0,\theta_0)d\Omega_0\) coincides with the quantity
\(A_0(\gamma_0,\theta_0)d\Omega_0\). In the case of a moving electron, however, the cross section \(\sigma(\gamma,\theta)d\Omega\) differs from the quantity \(A(\gamma,\theta)d\Omega\) by the factor
\(\dfrac{t_0}{t}=1-\dfrac{v}{c}\cos\alpha\), where \(\alpha\) is the angle between the direction of motion of the electron and the direction of propagation of the incident photons, and \(v\) is the velocity of the electron before collision with the photon. Indeed, photons crossing a stationary area

*) Since it is not obvious that the area of the transverse section of a light beam is invariant with respect to Lorentz transformations, we give here an elementary proof: the number of photons contained in a cylindrical volume of a light beam bounded by a section \(q\) and with generator length \(l=c\Delta t\) is invariant:

\[ N=\rho V=\rho'V', \tag{a} \]

where \(N\) is the total number of particles in the volume \(V\), and \(\rho\) is the photon density. Primed letters denote the corresponding quantities in the moving coordinate system. From (a) it follows that

\[ \rho c\Delta t q=\rho' c\Delta t' q'. \tag{b} \]

From (a) and (b) we obtain:

\[ \frac{V}{\Delta t q}=\frac{V'}{\Delta t' q'}. \tag{c} \]

Using the relation \(V\nu=V'\nu'\) \(^{17}\), from (c) we find:

\[ \nu q\Delta t=\nu' q'\Delta t'. \tag{d} \]

Since the number of wavelengths \(n\) fitting into the volume under consideration is invariant, \(\Delta t=nT\) and \(\Delta t'=nT'\), where \(T\) is the period corresponding to the frequency \(\nu\), formula (d) can be written in the form

\[ \nu Tq=\nu'T'q', \]

and consequently:

\[ q=q'. \]

during the time \(t_0\), will cross the platform moving together with the electron in a different time

\[ t=\frac{t_0}{1-\frac{v}{c}\cos\alpha}. \]

Thus, knowing the quantity \(A(\gamma,\theta)\), it is easy to find the scattering cross section on a moving electron:

\[ \sigma(\gamma,\theta)\,d\Omega=A(\gamma,\theta)\cdot\frac{t_0}{t}\,d\Omega . \]

According to Pauli, we find the quantity \(A(\gamma,\theta)\) from the condition

\[ A(\gamma,\theta)\,d\Omega=A_0(\gamma_0,\theta_0)\,d\Omega_0 . \]

Making the corresponding substitutions, we find the scattering cross section on an electron moving with velocity \(v\):

\[ \sigma(\gamma,\theta)\,d\Omega= \frac{r_0^2}{2}\, \frac{\gamma_1^2}{\gamma^2}\, \frac{(1-\beta^2)}{(1-\beta\cos\alpha)^2} \times \left[ \frac{\gamma}{\gamma_1}\cdot \frac{(1-\beta\cos\alpha)}{(1-\beta\cos\alpha')} + \frac{\gamma_1}{\gamma}\, \frac{(1-\beta\cos\alpha')}{(1-\beta\cos\alpha)} -\sin^2\theta_0 \right]d\Omega, \tag{30} \]

where \(\theta_0\) is the photon scattering angle in the reference system in which the electron is at rest before the collision, and

\[ \sin^2\theta_0= \sqrt{1-\beta^2} \left[ \frac{1}{\gamma_1(1-\beta\cos\alpha')} - \frac{1}{\gamma(1-\beta\cos\alpha)} \right]\times \]

\[ \times \left\{ 2-\sqrt{1-\beta^2} \left[ \frac{1}{\gamma_1(1-\beta\cos\alpha')} - \frac{1}{\gamma(1-\beta\cos\alpha)} \right] \right\}. \tag{31} \]

Here \(\gamma\) and \(\gamma_1\) are the energies of the incident and scattered photons, \(\alpha\) and \(\alpha'\) are the angles between the directions of the incident and, respectively, the scattered photons and the direction of the electron velocity before the collision, \(\beta=\dfrac{v}{c}\), and \(v\) is the electron velocity before the collision.

It is usually assumed that, when calculating the effective scattering cross section, one may neglect the motion of the electron before the act of scattering if the kinetic energy of the electron before scattering is much smaller than the energies of the incident and scattered photons. This, however, is not so. Independently of the energies of the incident and scattered photons, the scattering cross section on a moving electron differs from the scattering cross section on an electron at rest.

It follows from (30)—(31) that the motion of the electron may be neglected in calculating the Compton scattering cross section if the condi-

condition \(v \ll c\), i.e., if the kinetic energy of the electron is much smaller than \(mc^2\). This condition remains valid for arbitrarily large energies of the incident and scattered photons \(^{16}\). Already from this it is qualitatively clear that, even for very energetic incident and scattered photons, one cannot neglect the energy of the electron in the atom, treating the electron as free, if this energy is comparable with \(mc^2\). As shown in \(^{18}\), the deviations from the Klein–Nishina–Tamm formula will in this case be of the order of \(\left(\dfrac{v}{c}\right)^2\). For scattering by an electron of the \(K\)-shell this gives a deviation of the order

\[ \left(\frac{ze^2}{\hbar c}\right)^2 = \left(\frac{z}{137}\right)^2, \]

which is an appreciable quantity for heavy elements. Thus, in the scattering even of very energetic photons by electrons of the inner shells of heavy elements, the electrons can in no case be regarded as free. On the contrary, in the case of light elements one may practically always regard the electrons as free.

Using formulas (30) and (31), it is easy to obtain an expression for the Compton-scattering cross section in the system of the center of inertia of the incident photon and the electron, i.e., in the reference frame in which the total momentum of the photon and the electron is zero. The velocity of this system relative to the laboratory frame of reference can be obtained from the condition that the total momentum of the colliding particles in this system is zero and from the Lorentz transformations for momentum. A simple calculation leads to the following value of the velocity of the center-of-inertia system relative to the laboratory frame of reference:

\[ v=\frac{\gamma}{\gamma+1}\,c . \tag{32} \]

Substituting this expression into (30) and (31) in place of \(v\), we find the formula for the cross section in the center-of-inertia system of the colliding particles:

\[ \sigma_{\text{c.i.}}(\gamma,\theta)\,d\Omega_{\text{c.i.}} = \frac{r_0^2}{2} \left(\frac{2\gamma+1}{\gamma+1}\right) \left\{ \frac{1}{1+\gamma(1+\cos\theta_{\text{c.i.}})} + \right. \]

\[ \left. + \frac{1+\gamma(1+\cos\theta_{\text{c.i.}})}{(1+2\gamma)^2} - \frac{1-\cos^2\theta_{\text{c.i.}}} {\left[1+\gamma(1+\cos\theta_{\text{c.i.}})\right]^2} \right\} \,d\Omega_{\text{c.i.}}, \tag{33} \]

where \(\gamma\) expresses the energy of the incident photon in the laboratory frame of reference. The energy of the incident photon in the center-of-inertia system is expressed in terms of \(\gamma\) by the relation

\[ \gamma_{\text{c.i.}}=\frac{\gamma}{\sqrt{2\gamma+1}} . \tag{34} \]

Table II gives the values of \(d\sigma_{\mathrm{c.i.}}\) for various scattering angles \(\theta_{\mathrm{c.i.}}\) at \(\gamma \gg 1\).

Table II

Dependence of the differential cross section on the photon scattering angle \(\theta_{\mathrm{c.i.}}\) in the center-of-inertia system

\(\theta_{\mathrm{c.i.}},\) in degrees	\(d\sigma_{\mathrm{c.i.}}\)	\(\theta_{\mathrm{c.i.}},\) in degrees	\(d\sigma_{\mathrm{c.i.}}\)
0	\(r_0^2 \dfrac{1}{\gamma}\, d\Omega_{\mathrm{c.i.}}\)	120	\(r_0^2 \dfrac{1}{\gamma}\dfrac{17}{8}\, d\Omega_{\mathrm{c.i.}}\)
60	\(r_0^2 \dfrac{1}{\gamma}\dfrac{25}{24}\, d\Omega_{\mathrm{c.i.}}\)	180	\(r_0^2\, d\Omega_{\mathrm{c.i.}}\)
90	\(r_0^2 \dfrac{1}{\gamma}\dfrac{5}{4}\, d\Omega_{\mathrm{c.i.}}\)

Figure 7 shows the dependence of the differential cross section, referred to a unit solid angle in the center-of-inertia system of the colliding particles, on the scattering angle \(\theta_{\mathrm{c.i.}}\) in this same system for various energies of the incident photon. The energies of the incident photon are given for the laboratory frame of reference. Thus

Fig. 7. Dependence of the differential cross section per unit solid angle in the center-of-inertia system of the colliding particles on the angle \(\theta_{\mathrm{c.i.}}\) for various energies of incident photons.

the scattering cross section in the center-of-inertia system proves to be non-isotropic. At very high energies of the incident photon, almost all the scattered photons fly backward in a narrow cone. In this case the situation is as if the electron were a small “mirror” reflecting the incident photons in the backward direction. Of course, the total scattering cross section in the center-of-inertia system, obtained by integrating expression (33) over the solid angle \(4\pi\), is equal to the total scattering cross section (24)

in the laboratory coordinate system. Indeed, the effective scattering cross section can be regarded as an area perpendicular to the direction of motion of the incident photon, i.e., to the direction of the velocity of the center-of-inertia system. The size of an area perpendicular to the direction of the velocity of the new reference system is not changed under a Lorentz transformation, and, consequently, the total effective cross section is invariant with respect to this transformation.

c) Polarization in the Compton effect on oriented electrons

Usually the Compton-scattering cross section is calculated for the case of unoriented electrons; in calculating the cross section, summation is carried out over all possible spin directions in the initial and final states. Scattering by oriented electrons was considered theoretically in the works of Nishina \(^{19}\), Franz \(^{20}\), Fano \(^{21}\), and Ya. B. Zel’dovich \(^{22}\). As shown in Franz’s work \(^{20}\), the scattering cross section on oriented electrons and the polarization of the scattered radiation are different for different types of polarization of the incident photons. In the case of plane-polarized incident radiation, the scattering cross section does not depend on the orientation of the electron spin and is determined by the same formula (9) as for scattering by unoriented electrons. However, the character of the polarization of the scattered radiation depends on the orientation of the scattering electrons.

In the case of unoriented electrons, radiation scattered through large angles is unpolarized at high energies of the incident photons and partially polarized at incident-photon energies of the order of \(mc^{2}\). In the case of oriented electrons, the cross section is the same as for unoriented electrons, while the scattered radiation is circularly polarized at high energies and elliptically polarized at low energies \(^{20}\).

In the case of scattering of elliptically polarized radiation by oriented electrons, the magnitude of the effective cross section depends on the orientation of the spins of the scattering electrons. According to \(^{19}\), it is equal to:

\[ d\sigma=\frac{r_0^2}{2}\,d\Omega\,\frac{\gamma_1^2}{\gamma^2} \left\{ \frac{\gamma_1}{\gamma}+\frac{\gamma}{\gamma_1} -2\left[(\mathbf b_1\mathbf n_1)^2+(\mathbf b_2\mathbf n_1)^2\right] \right. \]
\[ \left. \pm 2b_1b_2(1-\cos\theta)\left[\gamma_1(\mathbf n\boldsymbol{\sigma})+\gamma\cos\theta(\mathbf n_0\boldsymbol{\sigma})\right] \right\}, \tag{35} \]

where the directions of the vectors \(\mathbf b_1\) and \(\mathbf b_2\) coincide with the directions of the principal axes of the polarization ellipse, and the lengths of these vectors are propor-

proportional to the lengths of the axes, and \(b_1^2+b_2^2=1\). The unit vectors \(\mathbf n\) and \(\mathbf n_1\) are directed along the incident and scattered rays, respectively; the unit vector \(\boldsymbol{\sigma}\) indicates the orientation of the electron spin before scattering. The sign \((+)\) corresponds to rotation of the plane of polarization of the incident radiation clockwise (i.e., right polarization), and the sign \((-)\) to left polarization. The scattered radiation is, generally speaking, also elliptically polarized.

The special case of scattering through \(180^\circ\) of circularly polarized radiation was considered by Ya. B. Zel’dovich\({}^{22}\). The same result can be obtained from the general formula. For incident radiation polarized in a right-handed circle (the index “п” in the cross section denotes right polarization),

\[ d\sigma_{\text{п},+}=r_0^2\,d\Omega\,\frac{1}{1+2\gamma}, \tag{36} \]

if \(\mathbf n\boldsymbol{\sigma}=1\), i.e., when the spin is oriented in the direction of the incident photon (index \(+\)),

\[ d\sigma_{\text{п},-}=r_0^2\,d\Omega\,\frac{1}{(1+2\gamma)^3}, \tag{37} \]

if \(\mathbf n\boldsymbol{\sigma}=-1\), i.e., for the opposite spin orientation. The symbols п, л, \(+\), \(-\) denote the right or left direction of rotation of the plane of polarization of the incident radiation and the orientation of the spin along \((+)\) or opposite to \((-)\) the direction of the incident photon. As is seen from (35), a simultaneous change in the spin orientation and in the direction of rotation of the plane of polarization of the incident photon does not change the magnitude of the scattering cross section. Therefore

\[ d\sigma_{\text{л},-}=d\sigma_{\text{п},+}\quad \text{and}\quad d\sigma_{\text{л},+}=d\sigma_{\text{п},-}. \tag{38} \]

As for the polarization of the scattered photons, right-polarized photons in back scattering give left-polarized photons, and conversely.

From (36), (37), and (38) it follows that

\[ \frac{d\sigma_{\text{п},+}}{d\sigma_{\text{л},+}}=(1+2\gamma)^2 . \tag{39} \]

The ratio of the back-scattering cross sections of right- and left-polarized photons on oriented electrons depends substantially on the energy of the incident photons. This difference in the cross sections can be detected experimentally in scattering by magnetized iron. In the case of complete circular polarization, for a single electron oriented on an atom, a change in the direction of magnetization of the iron changes the number of photons scattered backward by \(4\%\) at \(\gamma=\frac{1}{2}\), by \(7\%\) at \(\gamma=1\), and by \(8\%\) at higher energies\({}^{22}\).

SCATTERING OF PHOTONS OF DIFFERENT ENERGIES BY ELECTRONS

By measuring the number of electrons knocked forward out of a magnetized scatterer, and changing the direction of magnetization, one can determine the degree of circular polarization of the incident radiation.

Usually the scattering of unpolarized radiation is investigated. In this case the Klein–Nishina–Tamm formula is valid in its usual form (10). The scattered radiation is partially polarized. Therefore the cross section for repeated Compton scattering is no longer determined by formula (10). The cross section of repeated (double) Compton scattering is equal to \(^{20}\)

\[ \begin{aligned} d\sigma_{\text{double}} &= \frac{r_0^4}{4}\, d\Omega\, d\Omega' \left(\frac{\gamma_1'}{\gamma}\right)^2 \Bigg\{ \left(\frac{\gamma}{\gamma_1}+\frac{\gamma_1}{\gamma}-[\mathbf n \mathbf n_1]^2\right) \left(\frac{\gamma_1}{\gamma_1'}+\frac{\gamma_1'}{\gamma}-[\mathbf n \mathbf n_1]^2\right) \\ &\quad +2\bigl([\mathbf n \mathbf n_1][\mathbf n_1 \mathbf n_1']\bigr)^2 -[\mathbf n \mathbf n_1]^2[\mathbf n \mathbf n_1']^2 \\ &\quad +(1-\mathbf n\mathbf n_1)(1-\mathbf n_1\mathbf n_1') \bigl[\gamma(\mathbf n\boldsymbol\sigma_1)+ \gamma_1(\mathbf n\mathbf n_1)(\mathbf n\boldsymbol\sigma_1)\bigr] \bigl[\gamma_1'(\mathbf n_1'\boldsymbol\sigma_2)+ \gamma_1(\mathbf n\mathbf n_1')(\mathbf n_1\boldsymbol\sigma_2)\bigr] \Bigg\}, \end{aligned} \tag{40} \]

where \(\mathbf n\), \(\mathbf n_1\), \(\mathbf n_1'\) are unit vectors in the directions of the incident, primarily scattered, and secondarily scattered rays; \(\boldsymbol\sigma_1\) and \(\boldsymbol\sigma_2\) are unit vectors of the directions of the electron spins in the first and second scatterers; \(d\Omega\) and \(d\Omega'\) are elements of the solid angles in which the primary and secondary scattering occur. The quantities \(\gamma\), \(\gamma_1\), and \(\gamma_1'\) determine the energies of the incident, primarily scattered, and secondarily scattered photons, respectively, in units of \(mc^2\).

From (40) it is seen that the cross section of double scattering depends on the orientation of the electron spins in the first and second scatterers. The cross section will be greatest if

\[ \boldsymbol\sigma_1 \uparrow\uparrow \gamma \mathbf n+\gamma_1(\mathbf n\mathbf n_1)\mathbf n_1 \quad \text{and} \quad \boldsymbol\sigma_2 \uparrow\uparrow \gamma_1'\mathbf n_1' +\gamma_1(\mathbf n,\mathbf n_1')\mathbf n_1, \tag{41} \]

and smallest if one of the vectors \(\boldsymbol\sigma_1\) or \(\boldsymbol\sigma_2\) has the direction opposite to (41).

In the case of double Compton scattering at right angles, the cross section takes the form \(^{20}\):

\[ d\sigma_{\text{double}}^{90^\circ} = \frac{r_0^4}{4} \left(\frac{\gamma_1'}{\gamma}\right)^2 \left\{ 2(\mathbf n\mathbf n_1')^2 +\gamma\gamma_1' \left[2+\gamma_1^2+ (\mathbf n\boldsymbol\sigma_1)(\mathbf n_1'\boldsymbol\sigma_2) \right] \right\}. \tag{42} \]

The influence of the orientation of the electrons in the first and second scatterers is especially large if the directions of the incident ray and the secondarily scattered ray are mutually perpendicular, since in this case \(\mathbf n\mathbf n_1'=0\) and the cross section \(d\sigma_{\text{double}}^{90^\circ}\) is determined entirely by the second term in the braces. For energetic incident photons, the first term in the braces is always small in comparison with

second, and they may be neglected. The orientation of the electron spins in the scatterers continues to affect the magnitude of the scattering cross section, since even for large values of \(\gamma\) the value of \(\gamma_1\) remains of order unity for scattering through an angle of \(90^\circ\).

In the case of double scattering through an angle of \(180^\circ\), from (40) we obtain:

\[ d\sigma_{\text{double}}^{180^\circ} = \frac{r_0^4}{4}\,d\Omega\,d\Omega' \left(\frac{\gamma_1}{\gamma}\right)^2 \left\{ \left(\frac{\gamma}{\gamma_1}+\frac{\gamma_1}{\gamma}\right) \left(\frac{\gamma_1}{\gamma_1'}+\frac{\gamma_1'}{\gamma_1}\right) + \right. \]

\[ \left. + \left(\frac{\gamma}{\gamma_1}-\frac{\gamma_1}{\gamma}\right) \left(\frac{\gamma_1}{\gamma_1'}-\frac{\gamma_1'}{\gamma_1}\right) (\mathbf n\boldsymbol\sigma_1)(\mathbf n_1'\boldsymbol\sigma_2) \right\}. \tag{43} \]

From formula (42), in the case of double scattering through an angle of \(90^\circ\), \(\mathbf n_1' \perp \mathbf n\), we obtain the following value for the ratio of the maximum and minimum cross sections corresponding to different orientations of the spins of the scattering electrons:

\[ \frac{d\sigma_{\max}^{90^\circ}}{d\sigma_{\min}^{90^\circ}} = \frac{2+\gamma_1^2+|\boldsymbol\sigma_1|\,|\boldsymbol\sigma_2|} {2+\gamma_1^2-|\boldsymbol\sigma_1|\,|\boldsymbol\sigma_2|}. \tag{44} \]

In the case of magnetized iron, the number of electrons oriented along the direction of magnetization only slightly exceeds the number of electrons oriented opposite to the direction of magnetization. Under complete magnetization of iron, out of 26 electrons of the atom, on average only 2.4 electrons are oriented. Therefore, in (44) the absolute values \(|\boldsymbol\sigma_1|\) and \(|\boldsymbol\sigma_2|\), equal to unity (which indicates the orientation of the spins of all electrons in the direction of \(\boldsymbol\sigma_1\) and \(\boldsymbol\sigma_2\) in each of the scatterers), must be replaced by the values

\[ |\boldsymbol\sigma_1|=|\boldsymbol\sigma_2|=\frac{2.4}{26}. \]

In this case \(|\boldsymbol\sigma_1|\,|\boldsymbol\sigma_2|=0.0085\).

Consequently,

\[ \frac{d\sigma_{\max}-d\sigma_{\min}} {\frac12(d\sigma_{\max}+d\sigma_{\min})} = \frac{0.0085}{1+\frac12\gamma_1^2}. \tag{45} \]

The effect is, according to (45), \(0.85\%\) for soft primary incident radiation, and for hard incident radiation it is somewhat smaller. For double scattering through an angle of \(180^\circ\) (on magnetized iron scatterers)

\[ \frac{d\sigma_{\max}-d\sigma_{\min}} {\frac12(d\sigma_{\max}+d\sigma_{\min})} = 0.017\, \frac{ \left(\frac{\gamma}{\gamma_1}-\frac{\gamma_1}{\gamma}\right) \left(\frac{\gamma_1}{\gamma_1'}-\frac{\gamma_1'}{\gamma_1}\right) }{ \left(\frac{\gamma}{\gamma_1}+\frac{\gamma_1}{\gamma}\right) \left(\frac{\gamma_1}{\gamma_1'}+\frac{\gamma_1'}{\gamma_1}\right) }. \tag{46} \]

and, consequently, at high energies of the incident photons the effect amounts to 1%.

By measuring the intensity of the radiation in double scattering, one can judge the character of the polarization of the incident radiation.

d) Results of the new quantum electrodynamics

The formula for the Compton-scattering cross section used in the preceding sections was obtained on the basis of the Dirac equation by the methods of quantum electrodynamics with the use of perturbation theory. Quantum electrodynamics describes rather well a whole series of other elementary processes, but it contains difficulties which manifest themselves in the appearance of divergent expressions already in the second approximation of perturbation theory. In essence, the divergences arise because of the infinite self-energies of the particles participating in the interaction. The difficulties connected with the infinite self-energy of the electron had already arisen in classical electrodynamics. In quantum electrodynamics these difficulties not only did not disappear, but were aggravated by the appearance of a divergent self-energy of the electron caused by its interaction with the zero-point fluctuations of the electromagnetic field.

Many works by both Soviet and foreign authors have been devoted to the elimination of these theoretical difficulties[^11]. As is known, in recent times quantum electrodynamics has been substantially developed on the basis of the consistent implementation of the ideas of renormalization of charge and mass. The conclusions obtained in the theory can be compared with experimental data. Such experimental data, in agreement with the theory, are:

1. The shift of the energy levels of the electron in hydrogen-like atoms. The \(2S_{1/2}\)-level of the hydrogen atom is shifted by a small amount (of the order of \(1060\,Mc/s\)) in comparison with the position calculated according to the ordinary theory[^23].

2. The additional magnetic moment of the electron[^24]. In accordance with the theory, the ratio of the magnetic moment of the electron to the mechanical one proved to be equal to

\[ \mu=\frac{e\hbar}{2mc}\left(1+\frac{\alpha}{2\pi}\right) \]

(where \(\alpha=\dfrac{e^2}{\hbar c}\simeq \dfrac{1}{137}\), \(\hbar=\dfrac{h}{2\pi}\)) instead of the value \(\dfrac{e\hbar}{2mc}\) accepted in the old theory.

1. Still another experimental confirmation of the correctness of the new theory would be experiments making it possible to measure

effective cross sections of various processes with sufficiently high accuracy (\(\sim 1\%\)). The new theory gives expressions for the effective cross sections of various processes, differing from the usual ones by correction terms. The calculation of these radiative corrections has been carried out in a number of works by various authors. In particular, radiative corrections to Compton scattering were calculated in works \(^{25}\), but all of them were obtained only for the nonrelativistic case, while the work of Schafroth \(^{26}\), giving a radiative correction \(\sim 10\%\) to the formula for the Klein–Nishina–Tamm scattering cross section, proved to be incorrect. The most thorough work devoted to the calculation of radiative corrections to Compton scattering is Feynman’s work \(^{25}\).

Feynman \(^{25}\) considered the case of different energies of the incident photon and gave a summary table of calculation results for two scattering angles of the photon. The corrections (of order \(e^6\)) to the cross section determined by the Klein–Nishina–Tamm formula (of order \(e^4\)) are given by the relation

\[ d\sigma(\theta)=d\sigma_{\mathrm{K.-N.-T}}(\theta)\left(1+\frac{e^2}{\pi\hbar c}\delta\right), \tag{47} \]

where the value \(\delta\), as a function of the photon scattering angle, is shown in Fig. 8 for incident-photon energies \(2.62\ \mathrm{MeV}\) and \(17.6\ \mathrm{MeV}\) (in the laboratory coordinate system). These graphs indicate a minimum of the corrections for scattering angles near \(40^\circ\) and an increase of the corrections for angles close to \(0^\circ\). The dependence of the corrections to the Compton-scattering cross section on the energy of the incident photon is given in Table III for scattering angle \(\theta \simeq 0^\circ\). These corrections, as is evident—

Fig. 8. Dependence of Feynman’s correction \(\delta\) on the photon scattering angle \(\theta\) for different energies of the incident photons.

Table III

Corrections to the Compton-scattering cross section for photon scattering angles \(\theta \simeq 0^\circ\), \(\theta \simeq 180^\circ\) at different energies of the incident photons

Energy in the lab. system, in MeV	Correction in %, for \(\theta \simeq 0^\circ\)	Correction in %, for \(\theta \simeq 180^\circ\)
50	3.80	\(\sim 1\%\)
150	5.26	\(\sim 1\%\)
300	6.41	\(\sim 1\%\)
1000	8.80	\(\sim 1\%\)

but, as follows from the table, increase with energy. For a scattering angle \(\theta \simeq 180^\circ\), the corrections depend only weakly on the energy of the incident photon and amount to approximately \(1\%\) of the cross section determined by the Klein–Nishina–Tamm formula.

3. COMPTON SCATTERING ON VARIOUS ELEMENTARY PARTICLES

Up to now the effective cross section for Compton scattering on an electron has been considered. In this section effective cross sections are given for Compton scattering on various charged elementary particles differing in the values of their spin and magnetic moment.

In considering Compton scattering on the proton, it is also necessary to take into account the scattering of photons by the meson cloud of the proton.

a) Summary of the effective cross sections for photon scattering on charged particles possessing different spins and magnetic moments

The values of the cross sections for photon scattering on various elementary charged particles are given in the reviews by Pauli\(^{27}\) and V. L. Ginzburg\(^{28}\).

Table IV (on p. 626) gives the cross sections for various values of the spin and magnetic moment of the scattering particle (\(\mu\) is the particle mass). It is assumed that the scattering particle is at rest before the scattering process. The energies of the incident and scattered photons are \(K_0\) and \(K\), respectively.

The formulas for the cross section (3), (4) are applicable only for \(K_0 \ll 137\mu c^2\). Formulas (1), (2) are valid for arbitrary energies.

As is seen from the table, for spin and magnetic moment of the scattering particle equal to zero, the scattering cross section decreases with increasing photon energy \(K_0\), independently of the scattering angle \(\theta\).

In the case of a particle with spin \(\frac{1}{2}\) and magnetic moment 1, the Klein–Nishina–Tamm formula is valid, as investigated in detail in § 1. The scattering cross section in this case also decreases with increasing energy of the incident photon. For a particle with spin \(\frac{1}{2}\) and a magnetic moment different from unity, a cross section is obtained that increases with the energy of the incident photon. The same occurs when the spin and the magnetic moment of the particle are equal to unity. Such an increase at arbitrarily large energies would lead to inadmissible divergences of the cross sections; however, formulas (3) and (4) were obtained under the assumption \(K_0 \ll 137\mu c^2\).

Table IV

Cross-section formulas for particles with different spins and magnetic moments

	Spin in units of $\hbar$	Magnetic moment in units $x=\dfrac{e\hbar}{2\mu c}$	Scattering cross section at angle $\theta$	Total scattering cross section for $K\ll 1$
1	0	0	$\dfrac{r_0^2 K^2}{2K_0^2}\cos^2\theta\,d\Omega$	$\pi r_0^2\dfrac{\mu c^2}{K_0}$
2	$\dfrac{1}{2}$	1	$\dfrac{r_0^2 K^2}{2K_0^2}\left(\dfrac{K_0}{K}+\dfrac{K}{K_0}-\sin^2\theta\right)d\Omega$	$\pi r_0^2\dfrac{\mu c^2}{K_0}\left(\dfrac{1}{2}+\ln\dfrac{2K_0}{\mu c^2}\right)$
3	$\dfrac{1}{2}$	$x\ne 1$	$(x-1)^4\dfrac{r_0^2}{4}\dfrac{K}{K_0}\,d\Omega+\ldots$ $K\gg \mu c^2$	$\dfrac{\pi}{4}r_0^2(x-1)^4\dfrac{K_0}{\mu c^2}+\ldots$
4	1	1	$\dfrac{r_0^2 K^3}{2K_0^2}\left\{1+\cos^2\theta+\dfrac{1}{48(\mu c^2)^2}\left[KK_0(28-64\cos\theta+12\cos^2\theta)+(K^2+K_0^2)(29-16\cos\theta+\cos^2\theta)\right]\right\}d\Omega$	$\dfrac{5\pi}{18}r_0^2\dfrac{K_0}{\mu c^2}$

b) Scattering of photons by nucleons

In the scattering of high-energy photons by nucleons (protons or neutrons), apparently, the meson field of the nucleons plays an essential role. The proton is surrounded by a meson cloud of radius

\[ \sim \frac{\hbar}{\mu c}\simeq 10^{-13}\ \text{cm}, \]

where $\mu$ is the mass of the $\pi$-meson, and the effective cross section for the scattering of photons by such a complex particle may differ greatly from the effective cross section obtained from the Klein–Nishina–Tamm formula for a proton without the meson cloud belonging to it.

On the basis of the symmetric theory of scalar and pseudoscalar mesons, the cross section for the scattering of photons of various energies by nucleo-

nah was calculated by Foldy and Zakson.^29 For the case of the pseudoscalar variant of meson theory, the effective scattering cross section increases strongly at incident-photon energies close to the threshold for meson production.

Figure 9 gives the graph of the effective cross section as a function of energy, obtained by Foldy and Zakson.

As is seen from Fig. 9, there is a fairly large effect of resonance scattering at energies close to the threshold energy for meson production. This maximum of the total scattering cross section

Fig. 9. Dependence of the photon scattering cross section on nucleons on the energy of the incident photons, calculated by Foldy and Zakson.

Fig. 10. Curves of the angular distribution of scattered photons for different energies of incident photons in scattering on a nucleon, calculated by Foldy and Zakson.

of photons on protons, at photon energies of the order of 150 MeV, can be found experimentally. Figure 10 gives the curves of the angular distribution of scattered photons for different energies of the incident photons.

If there exist isobaric states of nucleons (states with higher values of spin and charge in comparison with the ground state^28), then for this reason, in scattering of a photon by a nucleon, a specific maximum should be observed in the region of energies corresponding to the energy of the excited isobaric state.

4. EXPERIMENTAL DATA AT LOW ENERGIES

Experimental studies in the region of low energies of incident photons have confirmed the correctness of the concepts of the photon and electron developed in quantum electrodynamics. The study of scattering at low energies of incident photons was carried out in the direction of studying the single act of scattering and in the direction of investigating the absorption of photons in matter.

At energies less than \(2mc^2\) it is easy to isolate the elementary act and determine the absorption coefficient in a substance due to Compton scattering. At energies exceeding \(2mc^2\), the experimental study of Compton scattering is made difficult by the fact that the pair-production effect begins to play an essential role, and at energies greater than \(4mc^2\) triplet production occurs (pair production in the field of an electron).

This paragraph describes work on the measurement of differential and integral cross sections and polarization effects in Compton scattering of photons with energies \(1\)—\(20\) MeV.

A major contribution to the study of Compton scattering was made by D. V. Skobeltsyn.

D. V. Skobeltsyn’s experiments\(^6\) on measuring the angular distribution of recoil electrons in Compton scattering, carried out in 1926—1929, showed the inadequacy of the Dirac–Gordon and Compton formulas for the scattering cross section. At the same time, D. V. Skobeltsyn’s detailed investigations directly confirmed the Klein–Nishina–Tamm formula. Thus, by these experiments the question of the fundamental equation of relativistic quantum mechanics was essentially resolved. In D. V. Skobeltsyn’s experiments the angular distribution of Compton electrons was studied in a Wilson chamber placed in a magnetic field. The measurements were performed with various sources of \(\gamma\)-rays and covered an energy range from \(160\) keV to \(\sim 3\) MeV.

Table V gives the results of measurements of the ratio of the number of recoil electrons for two angular intervals \(\left(\dfrac{N^{10^\circ}_{0^\circ}}{N^{20^\circ}_{10^\circ}}\right)\) as a function of the energy of the incident radiation.

Table V

Results of D. V. Skobeltsyn’s experiments

\(h\nu\) in keV	Observed \(\dfrac{N^{10^\circ}_{0^\circ}}{N^{20^\circ}_{10^\circ}}\)	Calculated according to Klein–Nishina	Calculated according to Compton	Calculated according to Dirac, Gordon
160—435	\(\dfrac{30}{49}=0.61\)	0.68	0.74	0.65
435—855	\(\dfrac{69}{89}=0.78\)	0.75	0.84	0.96
855—1300	\(\dfrac{58}{60}=0.97\)	0.93	0.95	1.01
1300—3000	\(\dfrac{117}{98}=1.2\)	1.15	1.03	0.94

The curves of the angular distribution for incident photon energies of 2.35 MeV are shown in Fig. 11. In the region of large scattering angles (from 60 to 90°) very good agreement is found between the experimental data and the theoretical data. This part of the distribution curve is a smooth curve, independent of the chosen angular interval in which the recoil electrons are counted.

If one takes the angular intervals 0—10°, 10—20°, 20—40°, 40—60°, 60—80° and 80—90° for the recoil electrons, then the curve will have the form shown in Fig. 12. Along the abscissa axis are plotted the angular intervals, and along the ordinate axis—the number of recoil electrons observed in the gas of the chamber in the selected angular interval. For comparison, a theoretical curve has been constructed, taking into account the size of the angular intervals. It is seen from the experiment that there is agreement with the Klein–Nishina curves, especially for incident photon energies of 2.35 MeV. For incident photon energies of 1.35 and 1.07 MeV, agreement is observed with the general course of the distribution curve. Thus, from D. V. Skobeltsyn’s experiments it was possible to conclude that the Klein–Nishina–Tamm formula is valid for photons of low energies.

Fig. 11. Curves of the angular distribution of recoil electrons. 1 — experiment, 2 — theory.

Fig. 12. Dependence of the number of recoil electrons on the angular intervals. The solid curve is theoretical, calculated by the Klein–Nishina–Tamm formula.

In the experiments of Crane, Gaerttner, and Turin^8 a Wilson chamber 15 cm in diameter and 2.5 cm deep was used, filled with air and ethyl-alcohol vapor at atmospheric pressure. The chamber was placed in a magnetic field. A scatterer and thin lead plates for absorbing the scattered photons were introduced into the chamber. Celluloid served as the scatterer, and in some—

in these experiments—mica. A beam of photons from a thorium source, previously collimated, entered and left the chamber through thin mica windows. The energy of the incident photons lay in the range from 0.5 MeV to 2.6 MeV. The setup is shown in Fig. 13. The emission of the recoil electron from the scatterer after the act of scattering at an angle \(\varphi\), and the emission of the scattered photon at an angle \(\theta\) to the direction of the incident photon, were recorded from the point of formation of the photoelectron in a thin lead plate. This method of selecting events made it possible to determine quite accurately both the angle \(\varphi\) and the angle \(\theta\). The relation between the measured emission angles of the recoil electron and of the scattered photon was compared with relation (6). From a sufficiently large statistical material (10,000 photographs,

Fig. 13. Setup for observing Compton scattering in a Wilson chamber: \(1\)—source; \(2\)—filter; \(3\)—shielding; \(4\)—photoelectron; \(5\)—target; \(6\)—recoil electron.

300 electron–photon coincidences), angular and energy relations were obtained for Compton scattering in this range of incident-photon energies. The experiments were carried out in 1936 and served as additional and vivid evidence for Compton’s relations, based on the laws of conservation of energy and momentum. Moreover, although the authors did not give detailed comparisons of the effective scattering cross section in the article, they concluded that the results of their measurements for the dependence of the effective cross section on the scattering angles of the electron and photon agree with calculations by the Klein–Nishina formula within the accuracy of the experiment. Unfortunately, the authors do not indicate the exact accuracy with which this agreement with theory was achieved.

The determination of the angular distribution of scattered \(\gamma\)-rays and recoil electrons using large scatterer thicknesses was carried out in the works of Kohlrausch \(^{32}\), Compton \(^{33}\), Chao \(^{34}\), and others. The difficulty of allowing for multiple scattering in the scatterer led to large errors in determining the angular distributions. When scatterers of small thickness are used

the number of scattering events is small. To improve the statistics, Bay and Schlesy³⁸ developed a rather original method for investigating scattered photons. A cross section of the apparatus for measuring the dependence of the Compton scattering cross section on the photon scattering angle is shown in Fig. 14. The layer of material scattering the photons (Al was used in the experiment) is arranged along an arc of a circle perpendicular to the plane of the drawing. The scattering angles corresponding to each individual element of the scatterer are equal to one another, as angles subtended by one and the same arc. Different scattering angles in the experiment were obtained by moving the scatterer through a certain distance from the line connecting the γ-ray source and the counter registering the scattered photons, and by changing the radius of the scatterer while keeping the distance between the source Ra and the counter Z unchanged.

Fig. 14. Schematic representation of the apparatus used by Schlesy.

The counter for registering the scattered photons and the γ-ray source were placed, as shown in Fig. 14, perpendicular to the plane of the drawing. The counter was shielded with lead from direct incidence of the primary γ-radiation. The counter readings were recorded by an amplifier with output to a mechanical register. The experiments were carried out for the range of photon scattering angles from 50 to 140°. The measurement results are given in Fig. 15. The experimental results were compared with the theoretical curve for the Compton scattering cross section, obtained taking into account the nonmonochromaticity of the incident radiation. The errors shown in Fig. 15 by dashed lines and having a rather considerable magnitude (~17%) are due to inaccuracies in determining the distance between Ra and Z, inaccuracies in determining the thickness of the scatterer, its height, the finite dimensions of the counter, etc.; but these errors do not strongly affect the general course of the curve. From the comparison of the experimental and theoretical curves one may conclude that, for angular

Fig. 15. Dependence of the Compton scattering cross section on the scattering angle, obtained in Schlesy’s experiments. The solid curve shows the theoretical dependence calculated by the Klein–Nishina–Tamm formula.

distributions of scattered photons, as well as from other experiments on the angular distribution of recoil electrons, there is agreement with the Klein–Nishina–Tamm formulas.

In connection with the development of counting techniques using scintillation counters and the development of counting circuits with high resolving power, in 1949–1950 a number of new experimental works appeared on the verification of angular and energy relations in Compton scattering, as well as on the measurement of the effective scattering cross section and polarization. The simultaneity of the appearance of the scattered photon and the recoil electron was also investigated. In the experiment of Hofstadter and Intaier36 the angular relations in scattering were investigated. The experimental arrangement is shown in Fig. 16. In the experiment coincidences between the recoil electron and the scattered photon were recorded. Depending on the position of the detector shown in Fig. 16, the change in the number of coincidences was studied.

Fig. 16. Scheme of the experiment of Hofstadter and Intaier: 1 — recoil-electron counters; 2 — scattered-photon counter.

The authors made corrections for detector efficiency and other factors, for example, absorption by the walls of the photomultiplier, and the change in the magnitude of absorption in the scattering crystal as a function of the photon energy and, consequently, of the angular position of the detector. The sum of the various corrections is given for different angles. For \(20^\circ\) it is \(9\%\), for \(50^\circ\)—\(21\%\), and for \(90^\circ\)—\(30\%\). The results of the experiment are shown in Fig. 17. The intensity of the \(\gamma\)-ray source \((\mathrm{Co}^{60})\) with energies \(1.69\) MeV and \(1.33\) MeV was determined with an accuracy of \(\sim 15\%\). As is evident from this work, the use of scintillation counters with some improvement in the experimental arrangement can lead to more accurate results, but still does not make it possible to measure the cross section with one-percent accuracy, even with good knowledge of the absolute intensity of the incident photons, since the indicated corrections will still be significant. The problem of simultaneity37 in Compton scattering was discussed for a long time. Do excited states of the electron exist and does the appear-

Fig. 17. Dependence of the number of counts on the detector position.

do the scattered photon and recoil electron appear simultaneously in a single act of Compton scattering? Modern theory gives for interaction times values far beyond experimental capabilities. (The interaction time is of order \(10^{-20}\) sec.)

Although there are no theoretical grounds for testing simultaneity, the authors, having at their disposal coincidence-counting equipment with a resolving time of \(10^{-8}\) sec., carried out experiments to prove directly that, in Compton scattering, the scattered photon and the recoil electron appear simultaneously within \(1.5\cdot 10^{-8}\) sec. Experiments to determine simultaneity were first performed by Bothe and Geiger\(^3\) in 1925. Simultaneity in their experiments meant the arrival together of the recoil electron and the scattered photon within a time interval of \(10^{-3}\) sec. These experiments were, in their time, extremely important, since they refuted the statistical theory of conservation laws\(^3\).

Hofstadter’s experiment\(^ {36}\) was carried out with a \(\gamma\)-ray source (\(\mathrm{Co}^{60}\)) by the method of scintillation counters operating in coincidence, with various prescribed delay times. Stilbene, naphthalene, and combined crystals were used. The experimental arrangement is shown in Fig. 16. Measurements were carried out for different angles of emission of the recoil electrons

Fig. 18. Schematic of the Cross and Ramsey apparatus

and showed (for any scattering angles) simultaneity of the appearance of the photon and electron within the resolving time \(1.5\cdot 10^{-8}\) sec.

In 1950 Cross and Ramsey\(^ {38}\) carried out experiments analogous to those of Bothe and Maier-Leibnitz\(^7\) with a \(\gamma\)-ray source from RaTh (energy \(2.62\) MeV). The apparatus is shown in Fig. 18. Unlike the preceding experiments, the different scattering angles \(\theta\) and \(\varphi\)

were set by moving the Be target to various distances from the source with the counter position fixed. The scattering Be target and the scintillation counters were enclosed in a helium-filled box with metal walls; the wall effect did not make a noticeable contribution to the number of counts, while replacing air by helium noticeably reduced the scattering of γ-rays and electrons in the gas. In processing the experimental results, corrections were introduced for the number of background coincidences associated with cosmic rays, radioactive contamination, etc. In the experiment the relation between the photon scattering angle and the recoil-electron emission angle was checked. The results are shown in Fig. 19. In order to verify that the directions of the incident beam, the scattered photon, and the recoil electron lie in one plane, the counter counting the recoil electrons was moved out of this plane by a distance of 2.5 cm. The number of coincidences for the normal arrangement of the counters was compared with the number of coincidences recorded by the setup when the electron counter was moved out of the plane. The results of this experiment are given in Table VI.

Fig. 19. Curve of the angular distribution of recoil electrons in the experiments of Cross and Ramsey.

Table VI

Difference in the number of coincidences as a function of the counter position

Position of the electron counter	Number of coincidences per hour	Number of coincidences per hour	Number of coincidences per hour
Position of the electron counter	without target	with target	effect
In the plane . . . . . . . .	14.3 ± 0.6	83.0 ± 1.4	68.7
2.5 cm above the plane . . .	7.3 ± 0.5	23.7 ± 1.0	16.4

In addition, measurements were carried out with the aim of a more accurate investigation of the maximum on the distribution curve of the number of recoil electrons as a function of the emission angle. The distribution curve is presented in Fig. 19. The solid curve denotes the theoretically calculated distribution with a maximum at 31.3°. The measured maximum lies, within the experimental error, in agreement with this value. The authors^38, unfortunately, did not carry out measurements of the angular distribution up to larger values of the angles, and we have no possibility of comparing their results

SCATTERING OF PHOTONS OF DIFFERENT ENERGIES BY ELECTRONS

with other measurements cited earlier. Further, the work showed that the expected number of electrons emitted in the direction predicted by the conservation laws appears simultaneously with the scattered photons within \(1.5\cdot 10^{-8}\) sec.*)

In the work of Del Sasso, Fowler, and Lauritsen \(^{39}\), using a Wilson chamber in a magnetic field, the scattering of \(\gamma\)-radiation with energy \(17.1\) MeV, obtained in the reaction \( \mathrm{Li}^{7} + \mathrm{H}^{1} \) in Pb and Al, was investigated. A collimated beam of \(\gamma\)-rays was passed through a chamber containing plates of the substance under study. In the Wilson chamber there were observed tracks of pairs formed in the plate, as well as tracks of single electrons and positrons. A considerable fraction of the single electrons and positrons belonged to pairs one of whose components was not registered either because of large losses of energy and scattering or because of imperfections in the photograph. The number of single electrons exceeded the number of positrons because of recoil electrons in Compton scattering in the substance under study. Table VII gives the numbers of pairs, single electrons, positrons, and recoil electrons in Pb and Al.

Table VII

Number of pairs, single electrons, positrons, and recoil electrons
in lead and aluminum

	Substance	Pairs	Electrons	Positrons	Recoil electrons
Without collimator . . . . . .	Pb	513	381	155
With collimator . . . . . .	Pb	257	101	49	52
With collimator . . . . . .	Al	71	105	12	93

The mean energy of the recoil electrons was estimated for incident photon energy \(17.1\) MeV. The mean energy of the recoil electrons proved to be \(12.7 \pm 0.7\) MeV, which is in agreement with the theoretical value of the mean energy, \(12.2\) MeV, calculated from the Klein–Nishina formula. In the experiment the energy distribution of the recoil electrons was also measured. The results of the experiment are pre-

*) More accurate measurements were made by Bell and Graham. They verified the simultaneity of the appearance of the scattered photon and the recoil electron within \(5\cdot 10^{-10}\) sec. \(^{38}\).

are shown in Fig. 20. The distribution obtained agrees quite well with theory.

In the work of Rosenblum, Schrader, and Werner^40 the total effective absorption cross sections in various substances (Cu, Sn, Pb, U) were measured for incident photon energies of 5.3; 10.3; and 17.6 MeV. The apparatus used in the experiments, analogous to that previously employed in the work of Lawson and De-Wire, is shown in Fig. 21. The Geiger counters were replaced by scintillation detectors. The source of γ-radiation was betatron radiation. The results of measurements of the effective cross sections for different energies of incident photons in various substances are given in Table VIII. A summary of the theoretical values of the effective cross sections for the photoeffect, Compton effect, pair production in the field of the nucleus and in the field of the electron, the nuclear photoeffect, and the total effective absorption cross section is given in the same Table VIII (on p. 637). The authors note that there is agreement between the results obtained and the results of Lawson^41 and De-Wire^42.

Fig. 20. Distribution of recoil electrons by energy.

Fig. 21. Schematic of the apparatus for measuring the total effective absorption cross section in various substances.

In the work of Berman^43, published in 1952, the total effective absorption cross section of photons with energy 19.5 MeV was measured. Measurements were carried out in various substances from hydrogen to uranium. Geiger counters served as detectors. By calculation from the total effective absorption cross section, ...

Table VIII

Results of measurements of effective cross sections for incident photon energies of 5.3 MeV and 17.6 MeV in various substances.
The second part of the table gives theoretical values of the effective cross sections for various processes.

Substance	Thickness (cm)	Density (g/cm³)	Absorption coefficient (cm⁻¹)	Effective absorption cross section (10⁻²⁴ cm²)
Experimental values of effective cross sections
a) 5.3 MeV
Cu	8,638	8,898	0,2735 ± 0,0027	3,244 ± 0,032
Sn	8,365	7,275	0,260 ± 0,004	7,05 ± 0,10
Pb	4,742	11,34	0,497 ± 0,005	15,07 ± 0,15
U	2,056	18,70	0,872 ± 0,009	18,43 ± 0,18
b) 17.6 MeV
Cu	7,817	8,878	0,3103 ± 0,0034	3,688 ± 0,040
Sn	8,365	7,275	0,3404 ± 0,0034	9,222 ± 0,092
Pb	3,005	11,34	0,6750 ± 0,0068	20,47 ± 0,21
U	2,000	18,70	1,198 ± 0,012	25,32 ± 0,25

Substance	Photoeffect cross section (10⁻²⁴ cm²)	Compton-effect cross section (10⁻²⁴ cm²)	Pair production in the nuclear field (10⁻²⁴ cm²)	Pair production in the electron field (10⁻²⁴ cm²)	Nuclear photoeffect (10⁻²⁴ cm²)	Total cross section (10⁻²⁴ cm²)
Theoretical effective cross sections
a) 5.3 MeV
Cu	0,0042	2,2822	0,9717	0,0115	≃ 0	3,2696
Sn	0,054	3,935	2,880	0,020	≃ 0	6,889
Pb	0,515	6,453	7,714	0,033	≃ 0	14,715
U	0,862	7,240	9,690	0,037	≃ 0	17,829
b) 17.6 MeV
Cu	0,0013	0,9691	2,496	0,059	0,14 ± 0,03	3,665
Sn	0,0174	1,6708	7,347	0,102	0,38 ± 0,07	9,517
Pb	0,168	2,7402	19,53	0,167	0,50 ± 0,10	23,11
U	0,281	3,074	24,51	0,187	0,56 ± 0,12	28,61

allocated the integral cross section of Compton scattering. The authors assert that the Klein–Nishina–Tamm formula is confirmed with an accuracy of \(\sim 7\%\).

In addition to the works mentioned, Compton scattering was also studied by a number of other investigators. The aggregate of measurements of integral and differential cross sections in various works, using diverse techniques, indicates agreement with the Klein–Nishina–Tamm formula within an accuracy of order \(10\%\) in the energy region of incident photons up to \(20\) MeV.

As is clear from § 2, in Compton scattering of low-energy photons the scattered radiation is partially polarized. A detailed experimental study of polarization in Compton scattering has been carried out only recently\(^{44}\). To study polarization it is most convenient to use double scattering.

The polarization of radiation in double Compton scattering was investigated experimentally in work\(^{45}\). As a source of \(\gamma\)-rays, \(\mathrm{Co}^{60}\) was used (intensity 5 curies). The apparatus employed is shown schematically in Fig. 22.

Fig. 22. Schematic of the apparatus for investigating the polarization of scattered photons in double Compton scattering.

The incident photon with momentum \(k_0\) was scattered by the copper scatterer \(S\) through an angle \(\theta_1\). The scattered photon, having momentum \(k_1\), was scattered a second time in the crystal of the scintillation counter \(C_1\), which registered the recoil electron in this scattering. The photon scattered in this secondary scattering event through an angle \(\theta_2\), upon entering the crystal \(C_2\) of the scintillation counter, was registered by means of the photoelectric effect. Counter \(C_2\) was shielded from photons scattered only once in the copper target by a lead layer \(5\) cm thick. The setup registered coincidences in counters \(C_1\) and \(C_2\), i.e., it registered double Compton scattering at specified angles \(\theta_1\) and \(\theta_2\) and at a specified azimuthal angle \(\varphi\) of the direction \(k_2\) relative to the plane \(k_0 k_1\).

The NaI crystals used had an area of \(2.5 \times 2.5\ \mathrm{cm}^2\) and were located at a distance of \(10\) cm from one another. Counter \(C_1\) stood at a distance of \(10\) cm from the copper target \(S\).

When the counters were rotated, their visible area remained unchanged. The azimuthal angle \(\varphi\) was varied in the measurements from \(0\)

to \(90^\circ\). The resolving time of the coincidence circuit was approximately \(10^{-7}\) sec.

Table IX gives the values of the ratio of the number of coincidences at zero azimuthal angle to their number at azimuthal angle \(\varphi\), for two values of the scattering angle \(\theta_1\), and for \(\theta_2=90^\circ\). The table also gives the theoretical values of this ratio, obtained with allowance for the solid angles subtended by the counter crystals. As can be seen from the table, the experimental and theoretical values are in general agreement.

Table IX

Ratio of the number of photons scattered at angle \(\varphi=0\) to the number of photons scattered at angle \(\varphi\), in double Compton scattering

| \multicolumn{3}{c}{Scattering angle \(\theta_1=83^\circ\)} | \multicolumn{3}{c}{Scattering angle \(\theta_1=50^\circ\)} |
|---:|---:|---:|---:|---:|---:|
| \(\varphi^\circ\) | experiment | theory | \(\varphi^\circ\) | experiment | theory |
| 90 | \(1.778 \pm 0.111\) | 1.76 | 90 | \(1.494 \pm 0.044\) | 1.445 |
| 70 | \(1.342 \pm 0.073\) | 1.561 | 70 | \(1.534 \pm 0.046\) | 1.368 |
| 50 | \(1.094 \pm 0.045\) | 1.218 | 50 | \(1.274 \pm 0.039\) | 1.222 |
| 30 | \(1.041 \pm 0.033\) | 1.085 | 30 | \(1.095 \pm 0.035\) | 1.081 |

The investigation of polarization thus indicates that the experimental data agree with the theoretical formulas presented in § 2. It is interesting to note that information on polarization in Compton scattering can be used to calculate polarization in other elementary processes, for example in pair production and bremsstrahlung. Such calculations were carried out by Mayem \(^{46}\) and Wick \(^{47}\). In the case of bremsstrahlung, the entire calculation is performed in the coordinate system in which the electron is at rest. The field of the nucleus moving in this system is represented as an aggregate of virtual photons with a spectrum \(1/\gamma\), where \(\gamma\) is the photon energy \(^{48}\). The calculation then considers the Compton scattering of these photons by an electron at rest. The scattered photons correspond to bremsstrahlung. After transformation to the laboratory frame of reference, effective cross sections for various polarizations of the emitted photons are obtained.

5. EXPERIMENTAL DATA AT HIGH ENERGIES

Until recently there were no experiments on the study of Compton scattering at high energies of the incident photons. If, in the region of low energies (of the order of \(1 \div 20\) Mev), one may state with confidence that the Klein–Nishina formula

... the Klein–Nishina formula is valid without substantial corrections, then for energies of the order of 100 MeV and above such a statement cannot be regarded as experimentally substantiated with sufficient reliability. Meanwhile, there are considerations that compel one to look for deviations from the usual form of the formulas for the scattering cross section in the region of sufficiently high energies. We have already mentioned in § 2 the corrections connected with the radiation reaction. These corrections are small and can hardly be detected with the existing accuracy of experiment. Somewhat larger are the radiative corrections to the effective scattering cross section obtained by Feynman^25. From Table III given in § 2 one can see the increase of these corrections with energy. Their experimental detection would be of considerable interest.

In addition to the corrections predicted theoretically in the as yet unexplored region of high energies, deviations from the known theoretical formulas for the effective cross sections of Compton scattering are also possible due to factors not taken into account by the existing theory. Thus, for example, the introduction in one way or another of an electron “structure” may lead to substantial changes in the formulas for the effective cross sections. In essence, the introduction of electron “dimensions” is dictated by the requirement that its self-energy be finite. For example, in order that in classical electrodynamics the energy of the electric field of the electron not diverge, it is necessary to ascribe finite dimensions to the source of the field (the electron). From the condition \(\dfrac{e^2}{r_0}\simeq mc^2\) there is determined the so-called classical radius of the electron \(r_0=10^{-13}\) cm. In an analogous way, for the “gravitational radius” of the electron one obtains the value \(r_{\mathrm{gr}}=10^{-55}\) cm. Taking into account the interaction of the electron with zero fluctuations of the electromagnetic field in vacuum leads to the introduction of a “dimension” \(\sim 10^{-70}\) cm. The calculation carried out by V. P. Silin^49 for one variant of the theory, taking into account the interaction of the electron with the meson field, gives the value of the electron “radius” \(\sim 10^{-16}\) cm^49.

Thus, from various considerations one may try to ascribe to the electron one or another size. Endowing the electron with definite dimensions, we have the right to expect deviations from the theoretical formulas for the effective scattering cross section obtained under the assumption of a point electron. Indeed, when the wavelength of the incident radiation is close to or smaller than the electron “dimensions,” effects must occur similar to the diffraction of a light wave of wavelength \(\lambda\) by an obstacle whose dimensions are greater than \(\lambda\). In other words, this may also be formulated as follows: in the case of high energy of the incident photon, most acts of Compton scattering are accompanied by the transfer of a significant momentum to the electron, i.e., in a large number

cases scattering occurs with a small “impact parameter.” For sufficiently small “impact parameters” the structure of the electron may make itself felt in an essential way.

All known attempts to introduce the size of the electron cannot yet be regarded as justified, but the theory of the point electron is not satisfactory either. The question of the structure of the electron thus remains open at the present time. Investigations in the region of high energies may be able to shed light on this most interesting question. Even a simple confirmation of the existing formulas of quantum electrodynamics for the effective cross section of Compton scattering in the high-energy region would in this respect be very important, since it would at least determine an upper limit for the assumed dimensions of the electron. Unfortunately, isolating the Compton effect in its pure form is a very difficult task, especially at high energy of the incident photon, since with increasing energy the Compton-scattering cross section decreases and becomes small in comparison with the cross sections of other processes, for example in comparison with the cross section for pair production. So far only one work has been carried out in which an attempt was made to test the Klein–Nishina–Tamm formula at a comparatively high (88 MeV) energy of the incident photons. In the experiments described^41 the accuracy of determining the cross section did not exceed 15%, and, moreover, the total integral scattering cross section was determined, which is due mainly to “impact parameters” still not small enough for one to expect the appearance of effects associated with the assumed structure of the electron. In order to isolate scattering events with small “impact parameters,” one should measure the differential cross section at large photon scattering angles, or else the integral cross section, but at considerably higher energies.

In addition to the already mentioned principal difficulty arising in the study of Compton scattering in the region of high energies (the smallness of the effective cross section in comparison with the cross sections of other processes), there is also another difficulty associated with obtaining photons of high energy. Experimenters do not have at their disposal a monochromatic (or at least nearly monochromatic) source of high-energy γ-radiation.

Even for comparatively small γ-radiation energies in the reaction Li^7 + H^1 there is a spreading in the energies of the emitted photons. At high γ-ray energies obtained from betatrons and synchrotrons, the γ-radiation has an energy spectrum. The spectrum of bremsstrahlung radiation is now sufficiently well known and is described by Schiff’s formula^50. When working with nonmonochromatic radiation it is impossible to determine the scattering cross section for a definite energy of the incident photons, and one has to, from the theo-

tical formulas, to compare the scattering cross section obtained for one or another energy interval of the incident photons. In this case the form of the spectrum enters into the theoretical formulas. In the works described, the measurement of the spectrum was not sufficiently accurate, as a result of which the measurement of the integral cross sections was also not entirely satisfactory.

Nor is it possible to measure the differential scattering cross section from recoil electrons, since for each scattering event the energy of the incident photon is unknown.

For high energies of the incident photons, at small “impact parameters” the recoil electrons fly out at small angles, the measurement of which with sufficient accuracy is very difficult. In addition to the difficulties indicated, with increasing energy other secondary effects have an ever greater influence (for example, meson production).

In Lawson’s experiments[^41] the absorption coefficients of high-energy \(\gamma\)-rays in various substances were measured. The absorption coefficients were measured with a statistical accuracy of \(1.5\%\).

In addition to the absorption coefficients, the number of electron-positron pairs formed in various substances was also measured directly. From a comparison of the data obtained, the author finds the complete effective cross section of Compton scattering in light substances. The experiments were carried out with the \(\gamma\)-radiation of the 100-MeV betatron. For measuring the absorption coefficient, the apparatus shown in Fig. 23 was used. The beam of \(\gamma\)-rays was passed through absorbers made of various substances: Be, Al, Cu, Sn, Pb, U.

Fig. 23. Diagram of Lawson’s experiments: 1 — absorber, 2 — cleaning magnetic field, 3 — target, 4 — counters, 5 — analyzing magnetic field, 6 — intensity meter, 7 — shielding, 8 — source of high-energy photons.

Absorbers specially examined for contamination were used. For all absorbers, except Be, the purity of the samples was not less than \(99.85\%\). After passing through the absorber, the beam was collimated by a lead collimator with two slits. The first slit cut out part of the beam; the second (wider) slit was calculated so that the beam did not touch its edges. On the path

from the first to the second slit the beam passed through a magnetic field, which cleaned the beam of charged particles. The second slit of the collimator did not transmit scattered particles. On leaving the collimator the beam struck a thin Au target, in which electron–positron pairs were formed. The electrons and positrons of the pairs, deflected by the magnetic field, were recorded by Geiger counters. Counts of individual groups of counters were recorded, as were coincidences of the counts of counters that counted electrons and of counters that counted positrons, i.e., electron–positron pairs were recorded; the simultaneous appearance of several pairs was also recorded. The triggering of the electronic circuits was synchronized with the emission of photons.

The pair spectrometer, consisting of several groups of counters, selected different pair energies in intervals of \(15\ \text{MeV}\). The entire apparatus, beginning with the cleaning magnetic field, was placed in a vacuum chamber in order to eliminate production in air along the path of the charged-particle beam. In the experiment, pairs with energy above a specified value, formed by photons above this energy, were selected. To determine the energy of the electrons and positrons of the pairs, a magnetic field of strength 4000 gauss was used. The diameter of the pole pieces was \(60\ \text{cm}\). The field was measured by the proton-resonance method\({}^{5)}\) with an accuracy up to \(0.01\%\), which is necessary for sufficiently accurate measurement of the energy of the particles of the pairs. By measuring the number of pairs with the spectrograph, it was possible to determine the change in the number of photons reaching the pair-spectrograph target as a function of the absorber material placed in the path of the beam. Thus, the number of pairs formed in the Au target served as an indicator of the number of photons with energy above a given value reaching this target. Since the authors were interested only in the high energies of the incident photons, only those pairs were selected whose energy exceeded approximately \(82\ \text{MeV}\). The mean effective energy recorded by the pair spectrograph was \(88 \pm 1\ \text{MeV}\). Because of fluctuations of the end of the spectrum, the value of the effective energy could vary. When the maximum energy changed by \(1\%\), the number of recorded pairs had to change by \(\sim 6\%\). Therefore, monitoring of the maximum energy of the emitted photons was necessary. The monitoring was carried out with an accuracy up to \(\sim 0.1\%\).

The absorption coefficients obtained and the corresponding absorption cross sections in various substances are given in Table X (p. 644). The absorption at these energies is due mainly to two processes:

1) formation of pairs in the field of the nucleus and in the field of the electron, and

2) Compton scattering. In Compton scattering, the photons scattered in the absorber partly leave the beam and are not registered by the pair spectrograph.

Table X

Absorption coefficients in various substances

Sample	cm²/g	cm²/atom	Statistical error in %
Be	0.0107	\(0.161\cdot 10^{-24}\)	1.2
Al	0.0252	\(1.128\cdot 10^{-24}\)	1.5
Cu	0.0471	\(4.971\cdot 10^{-24}\)	1.5
Sn	0.0665	\(13.11\cdot 10^{-24}\)	0.95
Pb	0.0909	\(31.27\cdot 10^{-24}\)	1.6
U	0.0973	\(38.46\cdot 10^{-24}\)	1.1

In order to separate the cross section of Compton scattering from the measured total absorption cross section, the following method is used.

The total absorption is measured in light and heavy substances—\(\sigma'\) and \(\sigma''\), respectively. The cross sections for pair production in these substances are determined experimentally—\(\sigma'_{\mathrm{p}}\) and \(\sigma''_{\mathrm{p}}\), respectively.

Let the cross section for Compton scattering be related to the cross section for pair production in the following way:

\[ \left. \begin{aligned} \sigma'_{\mathrm{k}} &= \alpha'\sigma'_{\mathrm{p}},\\ \sigma''_{\mathrm{k}} &= \alpha''\sigma''_{\mathrm{p}}, \end{aligned} \right\} \tag{48} \]

then

\[ \frac{\sigma'}{\sigma''} = \frac{\sigma'_{\mathrm{p}}(1+\alpha')}{\sigma''_{\mathrm{p}}(1+\alpha'')}. \tag{49} \]

For a substance with a large atomic number, the quantity \(\alpha''\) can be neglected in comparison with unity.

Then from (49)

\[ \frac{\sigma'}{\sigma''} = \frac{\sigma'_{\mathrm{p}}}{\sigma''_{\mathrm{p}}}(1+\alpha'), \tag{50} \]

whence one can determine the quantity \(\alpha'\), i.e. \(\sigma'_{\mathrm{k}}=\alpha'\sigma'_{\mathrm{p}}\). Thus, the Compton scattering cross section can be determined for a substance with a small atomic number.

The pair-production cross sections measured in the experiments differed from the theoretically calculated ones, which the author attributes to the inapplicability of the Born approximation in the case of heavy elements.

Therefore the total absorption cross sections found experimentally also turned out to differ from those calculated theoretically. In Fig. 24 the ratio of the theoretical values of the total absorption cross sections to the experimental ones is shown as a function of \(Z^2\). Since the corrections to the Born approximation are proportional to \(Z^2\), the points in Fig. 24 lie approximately on one straight line. The exception is the point corresponding to Be, for which the experimental value lies below this straight line. Table XI gives the theoretical values of the total cross sections for the processes of pair production in the nuclear field, pair production in the electron field, and Compton scattering. Comparison of the experimental data and the theoretical value of the Compton-scattering cross section (taking into account errors associated with false counts of pairs due to the presence of a diffuse background of scattered \(\gamma\)-rays and charged particles, multiple scattering in the absorber, and other similar effects) leads to the conclusion that the experimental value of the Compton-scattering cross section agrees with the Klein–Nishina–Tamm formula for incident photon energies of \(80\) MeV to within \(15\%\) accuracy.

Fig. 24. Ratio of the theoretical values of the total cross sections to the experimental ones as a function of \(Z^2\).

Fig. 25. Diagram of the De-Wire setup: 1 — source, 2 — collimator, 3 — absorber, 4 — part of the cleaning magnet field, 5 — shielding, 6 — shielding, 7 — target, 8 — Geiger counters placed in a magnetic field.

Table XI

Theoretical values of cross sections for various processes

	Be, cm²/g	Be, cm²/atom	Al, cm²/g	Al, cm²/atom	Cu, cm²/g	Cu, cm²/atom
Pair production in the nucleus	0.005928	0.0888·10⁻²⁴	0.02035	0.9113·10⁻²⁴	0.04186	4.418·10⁻²⁴
Pair production in the electron	0.001565	0.0234·10⁻²⁴	0.00167	0.0748·10⁻²⁴	0.00153	0.161·10⁻²⁴
Compton scattering	0.00240	0.0359·10⁻²⁴	0.00261	0.1169·10⁻²⁴	0.00247	0.261·10⁻²⁴
Total cross section	0.00989	0.1482·10⁻²⁴	0.02463	1.103·10⁻²⁴	0.04586	4.84·10⁻²⁴

	Sn, cm²/g	Sn, cm²/atom	Pb, cm²/g	Pb, cm²/atom	U, cm²/g	U, cm²/atom
Pair production in the nucleus	0.06511	12.83·10⁻²⁴	0.09803	33.72·10⁻²⁴	0.1068	42.21·10⁻²⁴
Pair production in the electron	0.00138	0.27·10⁻²⁴	0.00127	0.44·10⁻²⁴	0.0012	0.49·10⁻²⁴
Compton scattering	0.00228	0.45·10⁻²⁴	0.00214	0.74·10⁻²⁴	0.0021	0.83·10⁻²⁴
Total cross section	0.06877	13.55·10⁻²⁴	0.1014	34.9·10⁻²⁴	0.1101	43.53·10⁻²⁴

The measurement of the absorption cross section in various substances at an incident-photon energy of 280 MeV was carried out by DeWire[^42]. The apparatus used in these experiments is shown in Fig. 25. In its main features the apparatus is similar to Lawson’s apparatus described above[^41]. The absorption cross sections obtained in DeWire’s experiments[^42] are collected in Table XII.

Table XII

Effective absorption cross section of photons at an energy of 280 MeV

Absorber	Effective cross section, cm²/g	Statistical error, %
Be	0.01060	1.2
Al	0.0284	1.2
Cu	0.0521	1.5
Sn	0.0776	1.0
Pb	0.1059	1.2
U	0.1148	1.4

However, in the work cited no separation of the Compton-scattering cross section is made. By calculating the theoretical values of the pair-production cross sections in various substances, one can, using relation (50), determine the Compton-scattering cross section. In doing so it is necessary to use exact values of the pair-production cross sections; the Born approximation is not suitable.

Thus, on the basis of the data presented in the review, one may conclude that the formulas of quantum electrodynamics for the differential and integral cross sections of photon scattering by electrons are valid in the energy region up to 20 MeV. In the energy region up to 88 MeV there are experimental data confirming the correctness of the formula for the integral scattering cross section. At higher energies (photons from synchrotrons and photons in cosmic rays), the validity of the theoretical formulas of quantum electrodynamics for scattering can be judged on the basis of indirect data from measurements of the total absorption coefficient of photons in matter*). Since

) Note added in proof.
A short note has recently been published on work measuring the differential cross section of Compton scattering at an incident-photon energy of 250 MeV, carried out by the scintillation-counter method. The authors point to good agreement of the results of the measurements of the differential cross section with the Klein–Nishina–Tamm formula. (Bulletin of the American Physical Society 23*, 7 (1953).)

the Compton scattering cross section decreases with increasing photon energy and its role in the total absorption coefficient is not large; therefore, in order to observe any substantial change in the course of the absorption curves, considerable deviations from the Klein–Nishina–Tamm cross-section formula are necessary. Indirect data give no indication of such deviations even at very high energies.

In § 5 the possible influence of the structure of particles on photon scattering at high energies was indicated. Although it is now unclear in what way the structure of particles could enter into quantum electrodynamics, experiments on the scattering of high-energy photons are undoubtedly of interest.

The author expresses gratitude to L. A. Razorenov and Prof. V. L. Ginzburg for their help in preparing the review.

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